John Baez
Algebraic Stacks

Preface

This is a writeup of some material developed mainly by James Dolan, with help from me.

Definition and Examples

We’ll tentatively use the following new definition of the existing term ‘algebraic stack’:

An algebraic stack is a symmetric monoidal finitely cocomplete linear category.

Explanding this somewhat: an algebraic stack CC is, for starters, a symmetric monoidal kModk Mod-enriched category, where kModk Mod is the category of vector spaces over a fixed commutative ring kk. We also assume that CC has finite colimits, and that the operation of taking tensor product distributes over finite colimits.

There are more elegant ways to say the same thing, but let us forego them. More important is to notice that:

  • An algebraic stack can be seen as a categorified version of a commutative ring: the finite colimits play the role of ‘addition’, the tensor product plays the role of ‘multiplication, and the fact that this tensor product is symmetric plays the role of commutativity.

  • An algebraic stack can be seen as a kind of ‘theory’. This will have a moduli stack of models, in the more usual sense of the word ‘stack’ — and this is part of why the term ‘algebraic stack’ is justified.

  • The category of coherent sheaves on a stack, in the more usual sense of the word ‘stack’, is an example of an algebraic stack as defined above — and this is another justification for our terminology.

To see how the definition ties in to existing notions, it is good to consider some examples:

Example: Suppose XX is a projective algebraic variety over a field kk, and let CC be the category of coherent sheavesf kk-modules over XX. Then CC is an algebraic stack. The idea here is that we are thinking of CC as a kind of stand-in for XX. Indeed, for an affine algebraic variety XX we can use the commutative ring of algebraic functions as a stand-in for XX. For a projective variety there are not enough functions of this sort. However, there are plenty of coherent sheaves, and the category CC of such sheaves is a categorified version of a commutative ring.

Example: More generally suppose XX is a scheme over the commutative ring kk, and let CC be the category of coherent sheaves of kk-modules over XX. Then CC is an algebraic stack. (Need to check that this is really true: reference?)

Example: Let GG be an algebraic group over the field kk, and let CC be the category of finite-dimensional representations of GG. Then CC is an algebraic stack. Unlike the previous examples, this example is really ‘stacky’. In other words: instead of standing in for a set with extra structure, now CC is standing in for a groupoid with extra structure, namely the one-object groupoid GG.

Example: More generally, let GG be an group scheme over the commutative ring kk, and let CC be the category of representations of GG on finitely generated kk-modules. Then CC is again an algebraic stack. (Need to check that this is really true: reference?)

Example: More generally than all the examples above, let GG be an Artin stack over the commutative ring kk, and let CC be the category of coherent sheaves of kk-modules over XX. Then CC is an algebraic stack. (Need to check that this is really true: reference?)

Last revised on July 3, 2010 at 18:05:58. See the history of this page for a list of all contributions to it.